About Bianchi I with time varying constants

Abstract

In this paper we study how to attack through different techniques a perfect fluid Bianchi I model with variable G, and $\Lambda.$ These tactics are: Lie groups method (LM), imposing a particular symmetry, self-similarity (SS), matter collineations (MC). and kinematical self-similarity (KSS). We compare both tactics since they are quite similar (symmetry principles). We arrive to the conclusion that the LM is too restrictive and brings us to get only the flat FRW solution with G=const. and $\Lambda=0$. The SS, MC and KSS approaches bring us to obtain the following solution: G is a decreasing time function and $\Lambda\thickapprox t^{-2}$, with $\Lambda<0$, while the exponents of the scale factor must satisfy the conditions $\sum_{i=1}% ^{3}\alpha_{i}=1$ and $\sum_{i=1}^{3}\alpha_{i}^{2}<1,$ $\forall\omega \in(-1,1)$, relaxing in this way the Kasner conditions.